Pensieve header: Products of multiple exponentials in the β-yax algebra; continues BW.nb and pensieve://2017-06/; continued pensieve://2017-08/.

ME=MatrixExp; MF=MatrixForm;

Representing β-yax

y=(({{0, 0}, {-1, 0}})); a=(({{0, 0}, {0, -γ}})); x=(({{0, ((E^(-β))-1) γ}, {0, 0}})); b=(({{β, 0}, {0, 0}})); q=(E^β);
MF /@ {t=Simplify[β a-γ b],A=ME[-β a/γ], B=ME[-b], T=ME[t/γ]}

{a.x-x.a==γ x, x.A==q A.x, a.y-y.a==-γ y, b.y-y.b==-β y, x.y-q y.x==γ((({{1, 0}, {0, 1}}))-T.A.A)} // Simplify

P=(({{1, 0}, {0, 1}}));
Do[P=Expand[P.ME[(Subscript[η, i])y].ME[(Subscript[α, i])a].ME[(Subscript[ξ, i])x]],{i,2}];
P // MF

eqn0=Thread[Flatten/@(P==τo ME[ηo y].ME[αo a].ME[ξo x])]

Solve[eqn0[[1]],τo]

P=(({{1, 0}, {0, 1}}));
Do[P=Expand[P.ME[(Subscript[η, i])y].ME[(Subscript[α, i])a].ME[(Subscript[ξ, i])x]],{i,2}];
Column[
sol=First@Solve[Thread[Flatten/@(P==τo ME[ηo y].ME[αo a].ME[ξo x])], {τo,ηo,αo,ξo}] /. (v_->ℰ_) :> (v->Simplify@PowerExpand[ℰ])
]

Column[{fs=FullSimplify[#], TeXForm@fs}& /@ ({ηo,αo,ξo,τo,((Log[τo])/(-β γ))} /.sol)]

Series[((((E^β)+(-1+(E^β)) γ (Subscript[η, 2]) (Subscript[ξ, 1])))^(-1)),{β,0,3}]

Column@Series[{ηo,αo,ξo,τo,((Log[τo])/(-β γ))} /.sol, {β,0,3}]

Column[{fs=Collect[Normal@Series[#,{β,0,2}], β, FullSimplify], TeXForm@fs}& /@ ({ηo,αo,ξo,τo,((Log[τo])/(-β γ))} /.sol)]

P=(({{1, 0}, {0, 1}}));
Do[P=Expand[P.ME[(Subscript[η, i])y].ME[(Subscript[α, i])a].ME[(Subscript[ξ, i])x]],{i,3}];
Column[
First@Solve[Thread[Flatten/@(P==τo ME[ηo y].ME[αo a].ME[ξo x])], {τo,ηo,αo,ξo}]
]

xy to yax

Column[
sol1=First@Solve[Thread[Flatten/@(ME[ξ x].ME[η y]==τo ME[ηo y].ME[αo a].ME[ξo x])], {τo,ηo,αo,ξo}] /. (v_->ℰ_) :> (v->FullSimplify@PowerExpand[ℰ])
]

Column@Series[{ηo,αo,ξo,τo,((Log[τo])/(-β γ))} /.sol1, {β,0,2}]